Strange duality of weighted homogeneous polynomials

TL;DR

This paper explores a mirror symmetry between invertible weighted homogeneous polynomials in three variables, introducing duality concepts that extend Arnold's strange duality among singularities.

Contribution

It defines Dolgachev and Gabrielov numbers for these polynomials and establishes a duality that generalizes Arnold's strange duality.

Findings

Established a duality between weighted homogeneous polynomials and their duals.

Extended Arnold's strange duality to a broader class of singularities.

Provided new invariants (Dolgachev and Gabrielov numbers) for classifying these polynomials.

Abstract

We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold's strange duality between the 14 exceptional unimodal singularities.